%--------------------------
 % @Author: Jingqiao Hu
 % @Date: 2020-11-17 17:04:34
 % @LastEditTime: 2021-03-14 22:06:51
 
 % NOTE: The overall domain must be in [-a, a] & [-b, b]
 % The integration is:
 %  \int_{min_y}^{max_y} \int_{min_x}^{max_x} B(N(x, y)) dxdy
 %  N(x, y) =  1/4(1 - x/a)(1 - y/b)
 %  a = lx/2; b = ly/2
%--------------------------
% function B = shape_derivative_local2(coor, a, b)
%     GaussNodes = [-1/sqrt(3); 1/sqrt(3)]; GaussWeigh = [1 1];
%     L = [1 0 0 0; 0 0 0 1; 0 1 1 0];
%     
%     minx = coor(1,1); maxx = coor(1,2);
%     miny = coor(2,1); maxy = coor(2,2);
%     
%     B = cell(4,1);
%     
%     for i = 1:2
%         for j = 1:2
%             % projection to [-1, 1]
%             GN_x = (maxx+minx + abs(maxx-minx)*GaussNodes(i))/2; 
%             GN_y = (maxy+miny + abs(maxy-miny)*GaussNodes(j))/2; 
%             
%             dN_x = 1/4*[-(1-GN_y/b)  (1-GN_y/b) (1+GN_y/b) -(1+GN_y/b)]/a;
%             dN_y = 1/4*[-(1-GN_x/a) -(1+GN_x/a) (1+GN_x/a)  (1-GN_x/a)]/b;                
% 
%             dN(1,1:2:8) = dN_x; dN(2,1:2:8) = dN_y;            
%             dN(3,2:2:8) = dN_x; dN(4,2:2:8) = dN_y;   
% 
%             Be = L*dN;  
%             B{2*(i-1)+j} = GaussWeigh(i)*GaussWeigh(j)*Be;
%         end
%     end
% end

function gradN = shape_derivative_local(coor, a, b)    
    GaussNodes = [-1/sqrt(3); 1/sqrt(3)];
    gradN = cell(4,1);
    L = [1 0 0 0; 0 0 0 1; 0 1 1 0];

    minx = coor(1,1); maxx = coor(1,2);
    miny = coor(2,1); maxy = coor(2,2);

    for i = 1:2
        for j = 1:2
            gp = 2*(i-1)+j;       
            % GN_x = GaussNodes(i); 
            % GN_y = GaussNodes(j); 
            GN_x = (maxx+minx + abs(maxx-minx)*GaussNodes(i))/2; 
            GN_y = (maxy+miny + abs(maxy-miny)*GaussNodes(j))/2;            
            
            dN_x = 1/4*[-(1-GN_x)  (1-GN_x) (1+GN_x) -(1+GN_x)];            
            dN_y = 1/4*[-(1-GN_y) -(1+GN_y) (1+GN_y)  (1-GN_y)];   

            J = [dN_x; dN_y]*[ -a  a  a  -a;  
                               -b  -b  b  b]';          
            G = [inv(J) zeros(size(J)); 
                zeros(size(J)) inv(J)];  
               
            % dNdx = 3*8
            dN1(1,1:2:8) = dN_x; dN1(2,1:2:8) = dN_y;            
            dN1(3,2:2:8) = dN_x; dN1(4,2:2:8) = dN_y;   
            B2 = L*G*dN1; 
            
            gradN{gp} = B2; 
        end
    end
end